ANALYTICAL AND EXPERIMENTAL DETERMINALTION OF

NORMAL MODES OF VIBRATION OF STRING

Shubham YadavRohit KatariaShreya RajpalPranit Arora

MEL 314 LAB REPORT

PART 1

ANALYTICAL AND EXPERIMENTAL DETERMINALTION OF NORMAL MODES OF VIBRATION OF STRING

THEORY: As fundamental frequency of a string is given by this relation, f = [(T/µ) ^ (0.5)]/2L, fundamental frequency is proportional to square root of Tension or load here. And, it should also inversely decrease with length. And all harmonics should be integral multiple of this frequency.

EXPERIMENTAL SETUP:

L

Schematic representation of experimental setup

STRING

EXCITERWEIGHT

OBSERVATIONS:

a) Length = 120cm; weight = 200 g

frequency mode Node1 loc Node2 loc Node3 loc Node4 loc Node5 loc21 1 0 - - - 12042 2 0 - 60 - 12065 3 0 39.5 - 80.5 12084 4 0 31.5 61 91 120

b) Length = 120 cm; weight = 300 g

frequency mode Node1 loc Node2 loc Node3 loc Node4 loc24 1 0 - - 12048 2 0 61 - 12070 3 0 40.5 81 120

c) Length = 100 cm; weight = 200 g

frequency mode Node1 loc Node2 loc Node3 loc Node4 loc37 1 0 - - 10072 2 0 50 - 100108 3 0 33.5 66.5 100

d) Length = 90 cm; weight = 300 g

frequency mode Node1 loc Node2 loc Node3 loc Node4 loc34 1 0 - - 10068 2 0 50 - 100101 3 0 33 67 100

e) Length = 80 cm; weight = 200 g

frequency mode Node1 loc Node2 loc Node3 loc Node4 loc48 1 0 - - 8094 2 0 40 - 80140 3 0 26.5 52.5 80

f) Length = 80 cm; weight = 300 g

frequency mode Node1 loc Node2 loc Node3 loc Node4 loc38 1 0 - - 8076 2 0 40 - 80112 3 0 26.5 53 80

Theoretical values calculation:

a. For length = 120 cm: If only weight/tension is changed then, f/ [(T) ^0.5] should be constant. And assuming first fundamental frequency (i.e. for l = 120cm and w = 200g) is correct, fundamental frequency for w = 300 g is calculated. This is done for all cases of length.

But, if only length is changed then f*L should be constant and fundamental frequencies for different length can be calculated.

f ( w = 300 g, l = 120cm ) = 21*[(.3g) ^0.5]/ [(.2g) ^0.5] = 25.71

f ( w = 200 g, l = 100cm ) = 21*120/100 = 25.2

f ( w = 200 g, l = 80cm ) = 21*120/80 = 31.5

f ( w = 300 g, l = 100cm ) = 25.71*120/100 = 30.85

f ( w = 300 g, l = 80 cm ) = 25.71*120/80 = 38.565

Comparison between values obtained and theoretical values:

PARAMETERS Expected value Obtained value300,120 25.71 24200,100 25.2 37300,100 30.85 34200,80 31.5 48300,80 38.565 38

CONCLUSIONS:

1. Harmonics of a fundamental frequency are nearly integral multiple of fundamental frequency which shows that all frequencies are quite accurate considering limits of experiment and measurement done.

2. Considerable difference was observed between the theoretical values and experimental ones of fundamental frequencies. Some possible reasons for which are :a. String was not inextensible so mass per unit length was changing which resulted in deviation

from expected valuesb. Fundamental frequency for first set of parameters was treated as standard for calculation of

next ones, as former itself was not very reliable value, derived values were expected to have deviation.

c. High amplitude was observed over a period of frequency, there may have been error to note exact value.

3. Nodes were found to be at locations where they were expected to be or at least very near to it.

PART 2

SAND PATTERN FOR DIFFERENT HARMONICS

After Signal from function generator was applied to exciter with frequency as variable, and sand was sprinkled on its surface. Distinct pattern were observed for fundamental frequency and its harmonics.

Pattern were observed, different for different frequencies (different harmonics). Later harmonics were unable to show clear pattern.

Also, odd harmonics were more dominant in revealing pattern that even ones, that’s why photos of ony odd ones were clicked.

FUNDAMENTAL FREQUENCY: 210

1ST HARMONIC: 410

3RD HARMONIC: 650

4TH HARMONIC: 870

5TH HARMONIC: 1050